Finite Group Databasebeta

← #17958330

Subgroups — #17958330

Order 4352.

392 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544027C212yesmaximal
2#14543725C212yesmaximal
3#14543715C212yesmaximal
4#14531502C211yesmaximal, characteristic
5#11763825—22no—
6#11763654—22no—
7#11763654C2214yes—
8C2×#10587630C411yescharacteristic
9C2×Q8×Dic17C2211yescharacteristic
10C2×#10587573C411yescharacteristic
11#11762715C2211yescharacteristic
12#11762696—22no—
13#11762691—22no—
14#11762674C2211yescharacteristic
15#11762655C411yescharacteristic
16C17×#391C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×Dic34C2×C411yescharacteristic
19C2×C4⋊C4×C17C2×C411yescharacteristic
20C2×C42×C17C2×C411yescharacteristic
21Q8×Dic17—42no—
22#10587630—22no—
23#10587630C814yescomplemented
24C2×C4⋊Dic17D412yes—
25C2×C4⋊Dic17C2311yescharacteristic
26C2×#63225—22no—
27C2×#63225—21no—
28C2×C4×Dic17—21no—
29C2×C4×Dic17C2×C411yescharacteristic
30C2×C4×Dic17C2×C411yescharacteristic
31C2×C4×Dic17D412yes—
32#10587573—42no—
33C22⋊#3624—24no—
34C4⋊#3624—42no—
35C4×#3624—42no—
36C17×#95—22no—
37#10587502—21no—
38#10587502—21no—
39#10587502—22no—
40#10587502—22no—
41C2×Q8×C17—41no—
42C2×Q8×C17—41no—
43C2×Q8×C17C2×C812yes—
44C2×Q8×C17M1612yes—
45C22×C4×C17—21no—
46C22×C4×C17—21no—
47C22×C4×C17C22⋊C412yes—
48C22×C4×C17C22×C411yescharacteristic
49C22×Dic17—21no—
50C22×Dic17—21no—
51C22×Dic17C22⋊C411yescharacteristic
52C22×Dic17C22⋊C411yescharacteristic
53C22×Dic17C2×D412yes—
54C2×Dic34—21no—
55C2×Dic34—41no—
56C2×Dic34—41no—
57C2×Dic34C2×C812yes—
58C2×#3624—44no—
59C4⋊C4×C17—41no—
60C42×C17—41no—
61C4⋊Dic17—42no—
62C4⋊Dic17—41no—
63#63225—28no—
64#63225—42no—
65C4×Dic17—21no—
66C4×Dic17—22no—
67C4×Dic17—22no—
68C4×Dic17—22no—
69C4×Dic17—42no—
70C4×Dic17C2×C812yes—
71#11482—171nomaximal, Sylow
72C23×C17C2×C22⋊C411yescharacteristic
73Q8×C17—42no—
74Q8×C17—42no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—42no—
82C2×C4×C17—41no—
83C2×C4×C17—41no—
84C2×C4×C17C22⋊C814yes—
85C2×C4×C17C22×C811yescharacteristic
86C2×C4×C17C2×M1611yescharacteristic
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—41no—
92C2×Dic17—41no—
93C2×Dic17—41no—
94C2×Dic17—41no—
95C2×Dic17—41no—
96C2×Dic17—41no—
97C2×Dic17—42no—
98C2×Dic17#9912yes—
99Dic34—42no—
100Dic34—42no—
101#3624—84no—
102#2327—171no—
103#1461—172no—
104#1324—172no—
105#1319—172no—
106C22×C17—21no—
107C22×C17—21no—
108C22×C17C2×C22⋊C811yescharacteristic
109C22×C17C2×#9911yescharacteristic
110C22×C17#41311yescharacteristic
111C4×C17—21no—
112C4×C17—21no—
113C4×C17—24no—
114C4×C17—81no—
115C4×C17—81no—
116Dic17—22no—
117Dic17—42no—
118Dic17—42no—
119Dic17—41no—
120Dic17—42no—
121C2×C4⋊Q8—171no—
122C2×C4×Q8—171no—
123C2×C4×Q8—171no—
124#431—342no—
125#391—171no—
126#391#59511yescharacteristic
127#387—342no—
128#387—342no—
129#382—171no—
130#376—171no—
131#324—174no—
132#324—342no—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#145911yescharacteristic
136C2×C17#212911yescharacteristic
137C2×C17#213311yescharacteristic
138C22×Q8—171no—
139C22×Q8C2×#59511yescharacteristic
140C4⋊Q8—174no—
141C4⋊Q8—342no—
142C4×Q8—682no—
143C4×Q8—682no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—171no—
146C2×C4⋊C4—342no—
147C2×C4⋊C4—341no—
148C2×C4⋊C4C2×#59511yescharacteristic
149C2×C42—171no—
150C2×C42—172no—
151C2×C42—171no—
152C2×C42—341no—
153C2×C42C2×#59511yescharacteristic
154C4⋊C8—682no—
155C22⋊C8—344no—
156C4×C8—682no—
157#95—22no—
158#95—341no—
159#95—341no—
160#95—342no—
161#95—342no—
162C17#1148211yescharacteristic, complemented, Sylow
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—172no—
166C2×Q8—341no—
167C2×Q8—681no—
168C2×Q8—681no—
169C2×Q8#6324212yes—
170C2×Q8#6324312yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—171no—
174C22×C4—171no—
175C22×C4—172no—
176C22×C4—341no—
177C22×C4—341no—
178C22×C4C22⋊#59512yes—
179C22×C4C22×#59511yescharacteristic
180C2×C8—684no—
181C4⋊C4—41no—
182C4⋊C4—348no—
183C4⋊C4—681no—
184C4⋊C4—682no—
185C4⋊C4—682no—
186C42—41no—
187C42—172no—
188C42—342no—
189C42—342no—
190C42—342no—
191C42—341no—
192C42—682no—
193C23C2×C22⋊#59511yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—682no—
197Q8—682no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—172no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—681no—
213C2×C4—681no—
214C2×C4—681no—
215C2×C4—681no—
216C2×C4—681no—
217C2×C4—682no—
218C2×C4—681no—
219C2×C4#1058754314yes—
220C2×C4C2×#6324211yescharacteristic
221C2×C4C2×#6324311yescharacteristic
222C8—1364no—
223C22—21no—
224C22—21no—
225C22C2×#1058754311yescharacteristic
226C22D17⋊#9911yescharacteristic
227C22#1176379611yescharacteristic
228C4—21no—
229C4—21no—
230C4—24no—
231C4—81no—
232C4—81no—
233C4—342no—
234C4—682no—
235C4—682no—
236C4—682no—
237C4—681no—
238C2—21no—
239C2—21no—
240C2#1454402211yescharacteristic
241C2#1454506711yescharacteristic
242C2#1454508011yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenter#391#595
hypocenterC17#11482
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
23. #10587630C8semidirect
162. C17#11482semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145440272
#145437252
#145437152
#145315022
#1148217